We characterize topological conjugacy classes of one-sided topological Markov shifts in terms of the associated Cuntz–Krieger algebras and their gauge actions with potentials.

We show that a directed graph $E$ is a finite graph with no sinks if and only if, for each commutative unital ring $R$, the Leavitt path algebra $L_{R}(E)$ is isomorphic to an algebraic Cuntz–Krieger algebra if and only if the $C^{\ast }$-algebra $C^{\ast }(E)$ is unital and $\text{rank}(K_{0}(C^{\ast }(E)))=\text{rank}(K_{1}(C^{\ast }(E)))$. Let $k$ be a field and $k^{\times }$ be the group of units of $k$. When $\text{rank}(k^{\times })<\infty$, we show that the Leavitt path algebra $L_{k}(E)$ is isomorphic to an algebraic Cuntz–Krieger algebra if and only if $L_{k}(E)$ is unital and $\text{rank}(K_{1}(L_{k}(E)))=(\text{rank}(k^{\times })+1)\text{rank}(K_{0}(L_{k}(E)))$. We also show that any unital $k$-algebra which is Morita equivalent or stably isomorphic to an algebraic Cuntz–Krieger algebra, is isomorphic to an algebraic Cuntz–Krieger algebra. As a consequence, corners of algebraic Cuntz–Krieger algebras are algebraic Cuntz–Krieger algebras.

Results of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the k-graph associated with a given skeleton and collection of squares and show that two k-graphs are isomorphic if and only if there is an isomorphism of their skeletons which preserves commuting squares. We use this to prove directly that each k-graph Λ is isomorphic to the quotient of the path category of its skeleton by the equivalence relation determined by the commuting squares, and show that this extends to a homeomorphism of infinite-path spaces when the k-graph is row finite with no sources. We conclude with a short direct proof of the characterization, originally due to Robertson and Sims, of simplicity of the C*-algebra of a row-finite k-graph with no sources.

We consider the higher-rank graphs introduced by Kumjian and Pask as models for higher-rank Cuntz–Krieger algebras. Wedescribe a variant of the Cuntz–Krieger relations which applies to graphs with sources, and describe a local convexitycondition which characterizes the higher-rank graphs that admit a non-trivial Cuntz–Krieger family. We then proveversions of the uniqueness theorems and classifications of ideals for the $C^*$-algebras generated by Cuntz–Krieger families.

AMS 2000 Mathematics subject classification: Primary 46L05